Gap phenomenon of holomorphic maps between Shilov boundaries of type-I bounded symmetric domains

Abstract

Motivated by the gap phenomenon for proper holomorphic maps between complex unit balls, this paper investigates smooth CR maps $f$ from an open piece $M$ of the Shilov boundary of the unit ball $B^s$ into the Shilov boundary $S_{r^{\prime },s^{\prime }}$ of a higher-rank Type I bounded symmetric domain ${\Omega }_{r^{\prime },s^{\prime }}$. To the best of our knowledge, this is the first work to systematically establish a general gap phenomenon in the higher-rank setting. Our main result demonstrates that when the signature difference $s^{\prime }-r^{\prime }$ falls into the interval $k(s-1)\le s^{\prime }-r^{\prime }<(k+1)(s-1)$ for some integer $k$, the non-trivial component of $f$ is constrained to a much smaller Type I boundary. Up to automorphisms of the domain and target spaces, $f$ decomposes into the block-diagonal form $$f(z)=\left(\begin{array}{cc}{I}_{r^{\prime }-k}& 0\\ 0& \varphi (z)\end{array}\right),$$ where the essential component $\varphi :M\to S_{k,s^{\prime }-r^{\prime }+k}$ is a smooth CR map into the corresponding Shilov boundary. We provide explicit constructions showing that these dimensional bounds are sharp. As an immediate corollary, in the initial gap regime $s-1\le s^{\prime }-r^{\prime }<2s-2$, the map $f$ reduces to the standard linear embedding.